MinT - Best Known (88−33, 88, s)-Nets in Base 32

Best Known (88−33, 88, s)-Nets in Base 32

(88−33, 88, 324)-Net over F₃₂ — Constructive and digital

Digital (55, 88, 324)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (3, 11, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (3, 14, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)
3. digital (7, 23, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]
4. digital (7, 40, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(88−33, 88, 546)-Net in Base 32 — Constructive

(55, 88, 546)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. (12, 28, 258)-net in base 32, using
  - base change [i] based on (4, 20, 258)-net in base 128, using
    - 4 times m-reduction [i] based on (4, 24, 258)-net in base 128, using
      - base change [i] based on digital (1, 21, 258)-net over F₂₅₆, using
        net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
2. (27, 60, 288)-net in base 32, using
  - 3 times m-reduction [i] based on (27, 63, 288)-net in base 32, using
    - base change [i] based on digital (9, 45, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(88−33, 88, 5700)-Net over F₃₂ — Digital

Digital (55, 88, 5700)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(88−33, 88, large)-Net in Base 32 — Upper bound on s

There is no (55, 88, large)-net in base 32, because

31 times m-reduction [i] would yield (55, 57, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]